DOCSLIB.ORG

INTRODUCTORY NOTES on MODULES 1. Introduction One of the Most Basic Concepts in Linear Algebra Is Linear Combinations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
INTRODUCTORY NOTES on MODULES 1. Introduction One of the Most Basic Concepts in Linear Algebra Is Linear Combinations INTRODUCTORY NOTES ON MODULES KEITH CONRAD 1. Introduction One of the most basic concepts in linear algebra is linear combinations: of vectors, of polynomials, of functions, and so on. For example, the polynomial 7 − 2T + 3T 2 is a linear combination of 1;T , and T 2: 7 · 1 − 2 · T + 3 · T 2. The coefficients used for linear combinations in a vector space are in a field, but there are many places where we meet linear combinations with coefficients in a ring. p p Example 1.1. In the ring Z[ −5] let p be the ideal (2; 1 + −5), so by definition p p p p p p = 2x + (1 + −5)y : x; y 2 Z[ −5] = Z[ −5] · 2 + Z[ −5] · (1 + −5); p p which is the set of all linear combinations of 2 and 1 + −5 with coefficients in Z[ −5]. p Such linear combinations do not provide unique representation: for x and y in Z[ −5], p p p x · 2 + y · (1 + −5) = (x − (1 + −5)) · 2 + (y − 2) · (1 + −5): p So we can't treat x and y as \coordinates" of 2x + (1 + −5)y. In a vector space like Rn or Cn, whenever there is a duplication of representations with a spanning set we can remove a vector from p the spanning set, but in p this is not the case: if we could reduce the spanning set f2; 1 + −5g of p p p to a single element α, then p = Z[ −5]α = (α) would be a principal ideal in Z[ −5], but it can be shown that p is not a principal ideal. p p Instead of reducing the size of f2; 1 + −5g to get a nice spanning set for p with Z[ −5]- coefficients, we can try to do something else to standardize the representation of elements of p as linear combinations: restrict coefficients to Z. That works in this case because: p p p = 2x + (1 + −5)y : x; y 2 Z[ −5] p p p = 2(a + b −5) + (1 + −5)(c + d −5) : a; b; c; d 2 Z p p p = 2a + 2 −5b + (1 + −5)c + (−5 + −5)d : a; b; c; d 2 Z p p p = Z · 2 + Z · 2 −5 + Z · (1 + −5) + Z · (−5 + −5): Describing p in terms of linear combinations with Z-coefficients made the spanning set grow. We p can shrink the spanning set back to f2; 1 + −5g because the new members of this spanning set, p p p 2 −5 and −5 + −5, are Z-linear combinations of 2 and 1 + −5: p p p p 2 −5 = (−1)2 + 2(1 + −5) and − 5 + −5 = (−3)2 + (1 + −5): 1 2 KEITH CONRAD p p p Thus 2 −5 and −5 + −5 are in Z · 2 + Z · (1 + −5), so p p p = Z · 2 + Z · (1 + −5) = 2m + (1 + −5)n : m; n 2 Z : p p Using coefficients in Z rather than in Z[ −5], there is unique representation: if 2m+(1+ −5)n = p 2m0 + (1 + −5)n0 then p (2(m − m0) + (n − n0)) + (n − n0) −5 = 0: The real and imaginary parts are 0, so n = n0 and then m = m0. Thus, we can regard m and n as p \coordinates" for 2m + (1 + −5)n, and the situation looks a lot closer to ordinary linear algebra. p p Warning. That the set f2; 1 + −5g has its Z[ −5]-linear combinations coincide with its Z- p linear combinations is something of a fluke. In the ring Z[ d], the set of Z-linear combinations of p two elements generally does not coincide with the set of their Z[ d]-linear combinations. The use of linear combinations with coefficients coming from a ring rather than a field suggests the concept of \vector space over a ring," which for historical reasons is not called a vector space but instead a module. Definition 1.2. For a commutative ring R, an R-module M is an abelian group M on which R acts by additive maps respecting the ring structure of R when these maps are added and composed: there is a scalar multiplication function R × M ! M denoted by (r; m) 7! rm such that (1)1 m = m for all m 2 M. (2) r(m + m0) = rm + rm0 for all r 2 R and m; m0 2 M. (3)( r + r0)m = rm + r0m and (rr0)m = r(r0m) for all r; r0 2 R and m 2 M. In the special case when R = F is a field, an F -module is just an F -vector space by another name. Example 1.3. The set Rn of ordered n-tuples in R with componentwise addition and scalar multiplication by R as in linear algebra is an R-module. Example 1.4. Every ideal I in R is an R-module with addition and scalar multiplication being the operations in R. Example 1.5. A quotient ring R=I for an ideal I is an R-module where r(x mod I) := rx mod I for r 2 R and x 2 R. (It is easy to check that this is well-defined and satisfies the axioms.) So I and R=I are both R-modules, whereas in the language of ring theory, ideals and quotient rings are not the same kind of object: an ideal is almost never a ring, for instance. Example 1.6. The ring R[T ] is an R-module using obvious addition and scalar multiplication. Example 1.7. The set Map(R; R) of functions f : R ! R under pointwise addition of functions and the scalar multiplication (rf)(x) = r(f(x)) is an R-module. INTRODUCTORY NOTES ON MODULES 3 That addition in M is commutative actually follows from other axioms for R-modules: distribu- tivity both ways on (1 + 1)(m + m0) shows m + m + m0 + m0 = m + m0 + m + m0, and canceling the leftmost m's and rightmost m0's (addition on M is a group law) gives m + m0 = m0 + m for all m and m0 in M. We will show how many concepts from linear algebra (linear dependence/independence, basis, linear transformation, subspace) can be formulated for modules. When the scalars are not a field, we encounter genuinely new phenomena. For example, the intuitive idea of linear independence in a vector space as meaning no vector is a linear combination of the others is the wrong way to think about linear independence in modules. As another example, in a vector space each spanning set can be refined to a basis, but a module with a finite generating set does not have to have a basis (this is related to nonprincipal ideals). In the last section, we'll see how the concept of a module gives a new way to think about a question purely about matrices acting on vector spaces: for a field F and matrices A and B in Matn(F ), deciding if A and B are conjugate in Matn(F ) is the same as deciding if two F [T ]-module structures on F n (each one uses A or B) are isomorphic. 2. Basic definitions In linear algebra the concepts of linear combination, linear transformation, isomorphism, sub- space, and quotient space all make sense when the coefficients are in a ring, not just a field, so they can all be adapted to the setting of modules with no real changes. Definition 2.1. In an R-module M, an R-linear combination of elements m1; : : : ; mk 2 M is an element of M having the form r1m1 + ··· + rkmk where the ri's are in R. If every element of M is a linear combination of m1; : : : ; mk, we call fm1; : : : ; mkg a spanning set or generating set of M or say the mi's span (or generate) M. Linear combinations are the basic way to create new elements of a module from old ones, just as in linear algebra in Rn. For instance, an ideal (a; b; c) = Ra + Rb + Rc in R is nothing other than the set of R-linear combinations of a, b, and c in R. Example 2.2. We can view the ideal I = (1 + 2i) in Z[i] as both a Z[i]-module and as a Z-module in a natural way. As a Z[i]-module, we can get everywhere in I from 1+2i: I = Z[i](1+2i). As a Z- module, we can get everywhere in I from 1+2i and i(1+2i) = −2+i since I = Z(1+2i)+Zi(1+2i) = Z(1 + 2i) + Z(−2 + i). Mostly we will be interested in modules with finite spanning sets, but there are some important examples of modules that require infinite spanning sets. So let's give the general definition of spanning set, allowing infinite ones. 4 KEITH CONRAD Definition 2.3. A spanning set of an R-module M is a subset fmigi2I of M such that every m 2 M is a finite R-linear combination of the mi's: X m = rimi; i2I where ri 2 R for all i and ri = 0 for all but finitely many i. Notice in this definition that we require each linear combination of the mi's to have only finitely many nonzero coefficients. (Of course, if there are only finitely many mi's to begin with then this is no constraint at all.) In analysis, vector spaces may be equipped with a topology and we can talk about truly infinite linear combinations using a notion of convergence.
Load more

Recommended publications
  • Convolution.Pdf
    Convolution March 22, 2020 0.0.1 Setting up [23]: from bokeh.plotting import figure from bokeh.io import output_notebook, show,curdoc from bokeh.themes import Theme import numpy as np output_notebook() theme=Theme(json={}) curdoc().theme=theme 0.1 Convolution Convolution is a fundamental operation that appears in many different contexts in both pure and applied settings in mathematics. Polynomials: a simple case Let’s look at a simple case first. Consider the space of real valued polynomial functions P. If f 2 P, we have X1 i f(z) = aiz i=0 where ai = 0 for i sufficiently large. There is an obvious linear map a : P! c00(R) where c00(R) is the space of sequences that are eventually zero. i For each integer i, we have a projection map ai : c00(R) ! R so that ai(f) is the coefficient of z for f 2 P. Since P is a ring under addition and multiplication of functions, these operations must be reflected on the other side of the map a. 1 Clearly we can add sequences in c00(R) componentwise, and since addition of polynomials is also componentwise we have: a(f + g) = a(f) + a(g) What about multiplication? We know that X Xn X1 an(fg) = ar(f)as(g) = ar(f)an−r(g) = ar(f)an−r(g) r+s=n r=0 r=0 where we adopt the convention that ai(f) = 0 if i < 0. Given (a0; a1;:::) and (b0; b1;:::) in c00(R), define a ∗ b = (c0; c1;:::) where X1 cn = ajbn−j j=0 with the same convention that ai = bi = 0 if i < 0.
    [Show full text]
  • On Sandwich Theorem for Delta-Subadditive and Delta-Superadditive Mappings
    Results Math 72 (2017), 385–399 c 2016 The Author(s). This article is published with open access at Springerlink.com 1422-6383/17/010385-15 published online December 5, 2016 Results in Mathematics DOI 10.1007/s00025-016-0627-7 On Sandwich Theorem for Delta-Subadditive and Delta-Superadditive Mappings Andrzej Olbry´s Abstract. In the present paper, inspired by methods contained in Gajda and Kominek (Stud Math 100:25–38, 1991) we generalize the well known sandwich theorem for subadditive and superadditive functionals to the case of delta-subadditive and delta-superadditive mappings. As a conse- quence we obtain the classical Hyers–Ulam stability result for the Cauchy functional equation. We also consider the problem of supporting delta- subadditive maps by additive ones. Mathematics Subject Classification. 39B62, 39B72. Keywords. Functional inequality, additive mapping, delta-subadditive mapping, delta-superadditive mapping. 1. Introduction We denote by R, N the sets of all reals and positive integers, respectively, moreover, unless explicitly stated otherwise, (Y,·) denotes a real normed space and (S, ·) stands for not necessary commutative semigroup. We recall that a functional f : S → R is said to be subadditive if f(x · y) ≤ f(x)+f(y),x,y∈ S. A functional g : S → R is called superadditive if f := −g is subadditive or, equivalently, if g satisfies g(x)+g(y) ≤ g(x · y),x,y∈ S. If a : S → R is at the same time subadditive and superadditive then we say that it is additive, in this case a satisfies the Cauchy functional equation a(x · y)=a(x)+a(y),x,y∈ S.
    [Show full text]
  • Tensor Products of F-Algebras
    Mediterr. J. Math. (2017) 14:63 DOI 10.1007/s00009-017-0841-x c The Author(s) 2017 This article is published with open access at Springerlink.com Tensor Products of f-algebras G. J. H. M. Buskes and A. W. Wickstead Abstract. We construct the tensor product for f-algebras, including proving a universal property for it, and investigate how it preserves algebraic properties of the factors. Mathematics Subject Classification. 06A70. Keywords. f-algebras, Tensor products. 1. Introduction An f-algebra is a vector lattice A with an associative multiplication , such that 1. a, b ∈ A+ ⇒ ab∈ A+ 2. If a, b, c ∈ A+ with a ∧ b = 0 then (ac) ∧ b =(ca) ∧ b =0. We are only concerned in this paper with Archimedean f-algebras so hence- forth, all f-algebras and, indeed, all vector lattices are assumed to be Archimedean. In this paper, we construct the tensor product of two f-algebras A and B. To be precise, we show that there is a natural way in which the Frem- lin Archimedean vector lattice tensor product A⊗B can be endowed with an f-algebra multiplication. This tensor product has the expected univer- sal property, at least as far as order bounded algebra homomorphisms are concerned. We also show that, unlike the case for order theoretic properties, algebraic properties behave well under this product. Again, let us emphasise here that algebra homomorphisms are not assumed to map an identity to an identity. Our approach is very much representational, however, much that might annoy some purists. In §2, we derive the representations needed, which are very mild improvements on results already in the literature.
    [Show full text]
  • Notes on Elementary Linear Algebra
    Notes on Elementary Linear Algebra Adam Coffman Contents 1 Real vector spaces 2 2 Subspaces 6 3 Additive Functions and Linear Functions 8 4 Distance functions 10 5 Bilinear forms and sesquilinear forms 12 6 Inner Products 18 7 Orthogonal and unitary transformations for non-degenerate inner products 25 8 Orthogonal and unitary transformations for positive definite inner products 29 These Notes are compiled from classroom handouts for Math 351, Math 511, and Math 554 at IPFW. They are not self-contained, but supplement the required texts, [A], [FIS], and [HK]. 1 1 Real vector spaces Definition 1.1. Given a set V , and two operations + (addition) and · (scalar multiplication), V is a “real vector space” means that the operations have all of the following properties: 1. Closure under Addition: For any u ∈ V and v ∈ V , u + v ∈ V . 2. Associative Law for Addition: For any u ∈ V and v ∈ V and w ∈ V ,(u + v)+w = u +(v + w). 3. Existence of a Zero Element: There exists an element 0 ∈ V such that for any v ∈ V , v + 0 = v. 4. Existence of an Opposite: For each v ∈ V , there exists an element of V , called −v ∈ V , such that v +(−v)=0. 5. Closure under Scalar Multiplication: For any r ∈ R and v ∈ V , r · v ∈ V . 6. Associative Law for Scalar Multiplication: For any r, s ∈ R and v ∈ V ,(rs) · v = r · (s · v). 7. Scalar Multiplication Identity: For any v ∈ V ,1· v = v. 8. Distributive Law: For all r, s ∈ R and v ∈ V ,(r + s) · v =(r · v)+(s · v).
    [Show full text]
  • On the Injective Hulls of Semisimple Modules
    transactions of the american mathematical society Volume 155, Number 1, March 1971 ON THE INJECTIVE HULLS OF SEMISIMPLE MODULES BY JEFFREY LEVINEC) Abstract. Let R be a ring. Let T=@ieI E(R¡Mt) and rV=V\isI E(R/Mt), where each M¡ is a maximal right ideal and E(A) is the injective hull of A for any A-module A. We show the following: If R is (von Neumann) regular, E(T) = T iff {R/Mt}le, contains only a finite number of nonisomorphic simple modules, each of which occurs only a finite number of times, or if it occurs an infinite number of times, it is finite dimensional over its endomorphism ring. Let R be a ring such that every cyclic Ä-module contains a simple. Let {R/Mi]ie¡ be a family of pairwise nonisomorphic simples. Then E(@ts, E(RIMi)) = T~[¡eIE(R/M/). In the commutative regular case these conditions are equivalent. Let R be a commutative ring. Then every intersection of maximal ideals can be written as an irredundant intersection of maximal ideals iff every cyclic of the form Rlf^\te, Mi, where {Mt}te! is any collection of maximal ideals, contains a simple. We finally look at the relationship between a regular ring R with central idempotents and the Zariski topology on spec R. Introduction. Eckmann and Schopf [2] have shown the existence and unique- ness up to isomorphism of the injective hull of any module. The problem has been to take a specific module and find explicitly its injective hull.
    [Show full text]
  • Convolution Products of Monodiffric Functions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOC‘RNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 37, 271-287 (1972) Convolution Products of Monodiffric Functions GEORGE BERZSENYI Lamar University, Beaumont, Texas 77710 Submitted by R. J. Dufin As it has been repeatedly pointed out by mathematicians working in discrete function theory, it is both essential and difficult to find discrete analogs for the ordinary pointwise product operation of analytic functions of a complex variable. R. P. Isaacs [Ill, the initiator of monodiffric function theory, is to be credited with the first results in this direction. Later G. J. Kurowski [13] introduced some new monodiffric analogues of pointwise multiplication while the present author [2] succeeded in sharpening and generalizing Isaacs’ and Kurowski’s results. However, all of these attempts were mainly concerned with trying to preserve as many of the pointwise aspects as possible. Consequently, each of these operations proved to be less than satisfactory either from the algebraic or from the function theoretic viewpoint. In the present paper, a different approach is taken. Influenced by the suc- cess of R. J. Duffin and C. S. Duris [5] in a related area, we utilize the line integrals introduced in Ref. [l] and the conjoint line integral of Isaacs [l I] to define several convolution-type product operations. It is shown that if D is a suitably restricted discrete domain then monodiffricity is preserved by each of these operations. Further algebraic and function theoretic properties of these operations are investigated and some examples of their applicability are given.
    [Show full text]
  • UC Berkeley UC Berkeley Electronic Theses and Dissertations
    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title One-sided prime ideals in noncommutative algebra Permalink https://escholarship.org/uc/item/7ts6k0b7 Author Reyes, Manuel Lionel Publication Date 2010 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California One-sided prime ideals in noncommutative algebra by Manuel Lionel Reyes A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Tsit Yuen Lam, Chair Professor George Bergman Professor Koushik Sen Spring 2010 One-sided prime ideals in noncommutative algebra Copyright 2010 by Manuel Lionel Reyes 1 Abstract One-sided prime ideals in noncommutative algebra by Manuel Lionel Reyes Doctor of Philosophy in Mathematics University of California, Berkeley Professor Tsit Yuen Lam, Chair The goal of this dissertation is to provide noncommutative generalizations of the following theorems from commutative algebra: (Cohen's Theorem) every ideal of a commutative ring R is finitely generated if and only if every prime ideal of R is finitely generated, and (Kaplan- sky's Theorems) every ideal of R is principal if and only if every prime ideal of R is principal, if and only if R is noetherian and every maximal ideal of R is principal. We approach this problem by introducing certain families of right ideals in noncommutative rings, called right Oka families, generalizing previous work on commutative rings by T. Y. Lam and the author. As in the commutative case, we prove that the right Oka families in a ring R correspond bi- jectively to the classes of cyclic right R-modules that are closed under extensions.
    [Show full text]
  • Vector Space Theory
    Vector Space Theory A course for second year students by Robert Howlett typesetting by TEX Contents Copyright University of Sydney Chapter 1: Preliminaries 1 §1a Logic and commonSchool sense of Mathematics and Statistics 1 §1b Sets and functions 3 §1c Relations 7 §1d Fields 10 Chapter 2: Matrices, row vectors and column vectors 18 §2a Matrix operations 18 §2b Simultaneous equations 24 §2c Partial pivoting 29 §2d Elementary matrices 32 §2e Determinants 35 §2f Introduction to eigenvalues 38 Chapter 3: Introduction to vector spaces 49 §3a Linearity 49 §3b Vector axioms 52 §3c Trivial consequences of the axioms 61 §3d Subspaces 63 §3e Linear combinations 71 Chapter 4: The structure of abstract vector spaces 81 §4a Preliminary lemmas 81 §4b Basis theorems 85 §4c The Replacement Lemma 86 §4d Two properties of linear transformations 91 §4e Coordinates relative to a basis 93 Chapter 5: Inner Product Spaces 99 §5a The inner product axioms 99 §5b Orthogonal projection 106 §5c Orthogonal and unitary transformations 116 §5d Quadratic forms 121 iii Chapter 6: Relationships between spaces 129 §6a Isomorphism 129 §6b Direct sums Copyright University of Sydney 134 §6c Quotient spaces 139 §6d The dual spaceSchool of Mathematics and Statistics 142 Chapter 7: Matrices and Linear Transformations 148 §7a The matrix of a linear transformation 148 §7b Multiplication of transformations and matrices 153 §7c The Main Theorem on Linear Transformations 157 §7d Rank and nullity of matrices 161 Chapter 8: Permutations and determinants 171 §8a Permutations 171 §8b Determinants
    [Show full text]
  • Rigidity of Ext and Tor with Coefficients in Residue Fields of a Commutative Noetherian Ring
    RIGIDITY OF EXT AND TOR WITH COEFFICIENTS IN RESIDUE FIELDS OF A COMMUTATIVE NOETHERIAN RING LARS WINTHER CHRISTENSEN, SRIKANTH B. IYENGAR, AND THOMAS MARLEY Abstract. Let p be a prime ideal in a commutative noetherian ring R. It R is proved that if an R-module M satisfies Torn (k(p);M) = 0 for some n > R dim Rp, where k(p) is the residue field at p, then Tori (k(p);M) = 0 holds for ∗ all i > n. Similar rigidity results concerning ExtR(k(p);M) are proved, and applications to the theory of homological dimensions are explored. 1. Introduction Flatness and injectivity of modules over a commutative ring R are characterized by vanishing of (co)homological functors and such vanishing can be verified by testing on cyclic R-modules. We discuss the flat case first, and in mildly greater generality: For any R-module M and integer n > 0, one has flat dimR M < n if R and only if Torn (R=a;M) = 0 holds for all ideals a ⊆ R. When R is noetherian, and in this paper we assume that it is, it suffices to test on modules R=p where p varies over the prime ideals in R. If R is local with unique maximal ideal m, and M is finitely generated, then it is sufficient to consider one cyclic module, namely the residue field k := R=m. Even if R is not local and M is not finitely generated, finiteness of flat dimR M is characterized by vanishing of Tor with coefficients in fields, the residue fields R k(p) := Rp=pRp to be specific.
    [Show full text]
  • C*-Algebraic Schur Product Theorem, P\'{O} Lya-Szeg\H {O}-Rudin Question and Novak's Conjecture
    C*-ALGEBRAIC SCHUR PRODUCT THEOREM, POLYA-SZEG´ O-RUDIN˝ QUESTION AND NOVAK’S CONJECTURE K. MAHESH KRISHNA Department of Humanities and Basic Sciences Aditya College of Engineering and Technology Surampalem, East-Godavari Andhra Pradesh 533 437 India Email: [email protected] August 17, 2021 Abstract: Striking result of Vyb´ıral [Adv. Math. 2020] says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vyb´ıral used this result to prove the Novak’s conjecture. In this paper, we define Schur product of matrices over arbitrary C*- algebras and derive the results of Schur and Vyb´ıral. As an application, we state C*-algebraic version of Novak’s conjecture and solve it for commutative unital C*-algebras. We formulate P´olya-Szeg˝o-Rudin question for the C*-algebraic Schur product of positive matrices. Keywords: Schur/Hadamard product, Positive matrix, Hilbert C*-module, C*-algebra, Schur product theorem, P´olya-Szeg˝o-Rudin question, Novak’s conjecture. Mathematics Subject Classification (2020): 15B48, 46L05, 46L08. Contents 1. Introduction 1 2. C*-algebraic Schur product, Schur product theorem and P´olya-Szeg˝o-Rudin question 3 3. Lower bounds for C*-algebraic Schur product 7 4. C*-algebraic Novak’s conjecture 9 5. Acknowledgements 11 References 11 1. Introduction Given matrices A := [aj,k]1≤j,k≤n and B := [bj,k]1≤j,k≤n in the matrix ring Mn(K), where K = R or C, the Schur/Hadamard/pointwise product of A and B is defined as (1) A ◦ B := aj,kbj,k .
    [Show full text]
  • Equivariant Spectra and Mackey Functors
    Equivariant spectra and Mackey functors Jonathan Rubin August 15, 2019 Suppose G is a finite group, and let OrbG be the orbit category of G. By Elmendorf's theorem (cf. Talk 2.2), there is an equivalence between the homotopy theory of G-spaces, and the homotopy theory of topological presheaves over OrbG . In this note, we introduce Mackey functors and explain Guillou and May's version of Elmendorf's theorem for G-spectra. This result gives an algebraic perspective on equivariant spectra.1 1 Mackey functors Spectra are higher algebraic analogues of abelian groups. Thus, we begin by con- sidering the fixed points of G-modules, and then we turn to G-spectra in general. Suppose pM, `, 0q is a G-module. By neglect of structure, M is a G-set, and H G therefore its fixed points M – Set pG{H, Mq form a presheaf over OrbG . Spelled out, we have an inclusion MK Ðâ MH for every inclusion K Ă H Ă G of subgroups, ´1 and an isomorphism gp´q : MH Ñ MgHg for every element g P G and subgroup H Ă G. The extra additive structure on M induces extra additive structure on the fixed points of M. Suppose H Ă G is a subgroup. By adjunction, an H-fixed point G{H Ñ M is equivalent to a G-linear map ZrG{Hs Ñ M, and thus MH – homZrGspZrG{Hs, Mq is an abelian group. More interestingly, for any inclusion of subgroups K Ă H Ă G, there is a G-linear map ZrG{Hs Ñ ZrG{Ks that rep- H resents the element H{K rK P ZrG{Ks .
    [Show full text]
  • Cyclic and Multiplication P-Bezout Modules
    International Journal of Algebra, Vol. 6, 2012, no. 23, 1117 - 1120 Cyclic and Multiplication P -Be´zout Modules Muhamad Ali Misri Department of Mathematics Education, IAIN Syekh Nurjati Cirebon Jl. By Pass Perjuangan Kesambi, Cirebon, Indonesia [email protected], [email protected] Irawati Department of Mathematics, Institut Teknologi Bandung Jl. Ganesa 10 Bandung 40132, Indonesia [email protected], [email protected] Hanni Garminia Y Department of Mathematics, Institut Teknologi Bandung Jl. Ganesa 10 Bandung 40132, Indonesia [email protected] Abstract We study P -B´ezout, Noetherian and multiplication modules. We inves- tigate the sufficient condition of a module over P -B´ezout ring is to be P -B´ezout module and even for arbitrary ring. Let R be a commutative ring with an identity and M be a cyclic multiplication R-module which has some properties of its submodule, we prove that M is P -B´ezout module. Mathemaics Subject Classification: 13C05,13C99 Keywords: P -b´ezout module, finitely generated submodule, prime sub- module, cyclic submodule 1 Introduction The ring which considered in this paper is commutative with an identity and will be denoted by R. The concept of P -B´ezout ring was introduced and studied in Bakkari [1] as a generalization of the B´ezout ring. A ring R is said to be P -B´ezout if every finitely generated prime ideal I over R is principle. 1118 M. A. Misri, Irawati and H. Garminia Y We will adapt that concept into module. A module M over R is said to be P -B´ezout if every finitely generated prime submodule N of M is cyclic.
    [Show full text]